Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification

Abstract

In this paper, the Boundary Element Method (BEM) for 3-D elastostatic problems is studied for the analysis of shell or shell-like structures. It is shown that the conventional boundary integral equation (CBIE) for 3-D elasticity does not degenerate when applied to shell-like structures, contrary to the case when it is applied to crack-like problems where it does degenerate due to the closeness of the two crack surfaces. The treatment of the nearly singular integrals, which is a crucial step in the applications of BIEs to thin shapes, is presented in detail. To verify the theory, numerical examples of spherical and ellipsoidal vessels are presented using the BEM approach developed in this paper. It is found that the system of equations using the CBIE is well conditioned for all the thickness studied for the vessels. The advantages, disadvantages and potential applications of the proposed BEM approach to shell-like structures, as compared with the FEM regarding modelling and accuracy, are discussed in the last section. Applications of this BEM approach to shell-like structures with non-uniform thickness, stiffeners and layers will be reported in a subsequent paper.

Continuum Models of Carbon Nanotube-Based Composites Using the Boundary Element Method

“…(1) is a weakly-singular form of the conventional BIE and does not involve computations of any singular integrals in the discretization. This BIE can be applied safely to thin shelllike materials or structures, as long as the nearly-singular integrals can be computed accurately and efficiently [27,31].…”

“…For example, the integration may be on the outer surface of a CNT (a thin cylindrical shell in the continuum model), while the source point is located on the inner surface of the CNT. Although other methods such as subdivisions can be applied to deal with these nearly-singular integrals, the most efficient and accurate one is believed to be the line-integral approach [27,72]. For example, the integral with the traction kernel on a surface S ∆ with the source point 0 P nearby can be isolated and further transformed into line integrals using the Stokes' theorem as follows [27,72]:…”

“…This solid angle integral (a surface integral) can be transformed into a line integral on the bounding curve C as well [27]:…”

“…Besides the often-cited advantage of the reduction in dimensions of the computational models, the BIE/BEM is superior in stress analysis, especially at the material interfaces, which are crucial to understanding the performance and integrity of materials. In recent years, the BEM has been found especially efficient in modeling many materials involving thin shell-like structures [27] or small features, such as composite materials with interphases (see, e.g., Refs. [28][29][30]), thin elastic films and coatings [31][32][33], thin piezoelectric films [34,35], and the micro-electro-mechanical systems (MEMS) [36][37][38][39][40].…”

Regularization of nearly singular integrals in the boundary element analysis for interior anisotropic thermal field near the boundary

“…Huang and Cruse (1993) presented another approach taking a coordinate transformation to relax nearly singular kernels. Other approaches include Gaussian integration with fine subdivisions, kernel cancellation methods (Nakagawa, 1993), the auxiliary surface of "tent" method (Lutz et al, 1992), and the line integral method (Krishnasamy et al, 1994, Liu et al, 1993, and Liu, 1998. Sladek et al (2001) proposed a semi-analytical integration scheme to deal with the logarithmic singularity in BEM.…”